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TopologyTopology OptimizationOptimization OfOf A A StampingStamping DieDieAnnaAnna NilssonNilsson andand FridaFrida BirathBirathAnna Nilsson, Altair Engineering AB, Ideon Research Park, Scheelevgen15, 223 70 Lund, SwedenFrida Birath, Epsilon HighTech Innovation AB, Telegrafgatan 8A, 169 84Stockholm, SwedenAbstract.Abstract. Today the casting structure of stamping tools is dimensionedaccording to standard guidelines. The main task is to develop analternative method of manufacturing stamping tools, which takes thestructural response of the tool into account to produce a lightweightdesign. This is done by performing a topology optimization with theobjective to improve the structural stiffness and to reduce the weightof the tool. The die is the only part of the stamping tool considered inthe methodology generation. The method established can be applied to anyother part of the stamping tool analogously, after the load cases of thepart are determined. Two different load cases are applied to the die tool,one operation case and one transportation case, which are weighted equallywhen optimizing. A forming simulation is made using LS Dyna to obtain theload on the die during operation. The attained loads are, together withthe loads affecting the die during transportation, applied to the die anda topology optimization is performed, using the software OptiStruct. ACAD model is set up from the result of the topology optimization. ThisCAD model is analysed and compared to the original die, i.e. the die thatis in use today. When comparing the result of the analyses, the maximaldisplacement, the von Mises stresses and the weight of the die isconsidered. The final results show an overall more homogenous stressdistribution, a reduction of the maximal displacement with around 15 %and a weight reduction of about 20 %.Keywords:Keywords: Finite Element Method, Topology Optimization, Sheet MetalForming.INTRODUCTIONINTRODUCTIONVolvo Cars Body Components (from now on VCBC), has great experience intool design and sheet metal forming and have been producing bodycomponents for Volvo cars since the 1920s. The experience gained hashelped to improve the design of the tool over the years. Today, the castingstructure of the tool is dimensioned according to standard guidelines 1.These guidelines are based on trial-anderror and not on a structuraloptimization. The development has also lead to use of body componentmaterials of greater and higher strength, which calls for a greater forceto be used in the forming process. More material has traditionally beenapplied to the stamping tool for it to sustain this increased load, andin order to reduce the risk of a large structural response. However, thiscourse of action cannot carry on indefinitely, since the lifting capacityof traverse cranes, needed for transporting the tools in the plant, islimitedandsoonreached.Thereforeanalternativemethodofmanufacturing tools, which takes the structural response of the tool intoaccount to produce a lightweight design, is called for.METHODMETHODA method for a manufacturing process of stamping tools, which takes thestructural response of the tool into account, while reducing its weighthas been generated. The method is presented in the flowchart in Figure1. It is applied to the tool forming the Volvo S80 truck lid. Only onepart of the tool, the die, is being optimized. The optimized form of theother tool parts may be derived analogously. See Figure 2. The method isbased on Finite element software HyperWorks, using the linear solverOptistruct. Two different load cases are applied to the die tool, oneoperation case and one transportation case, which are weighted equallyduring optimization. The operation case is attained from a formingsimulation using LS Dyna, where contact pressure and drawbead forces fromthe last time step of the simulation on the die are found and later onapplied during topology optimization. The second load case is the liftingof the die in the lifting lugs, an operation needed for transportationand cleaning. In the set up of the topology optimization problem, the twoload cases are weighted equally. The forming load case occurs to a muchlarger extent than the lifting case, however, as a safety precaution theyare still given the same relevance.SheetSheet MetalMetal FormingForming SimulationSimulationThe loads acting on the die during operation need to be known in orderto be able to perform a topology optimization. The loads from themanufacturing process are obtained through a sheet forming simulation.A finite element model is created from the CAD geometry of the tool. Thismodel consists of the same parts as the original tool. However, the punch,blankholder and die are approximated to be rigid, and therefore only thesurfaces in contact with the blank are needed in the model, see Figure3. For details in the forming simulation performed see 2. The contactpressure and drawbead forces found are mapped from the LS-Dyna model tothe finite element model of the topology optimization, by use of a modifiedDelaunay algorithm 2.TopologyTopology OptimizationOptimizationThe method is based on determining the optimal distribution of thematerial within the die, to attain an as stiff structure as possible. Toachieve this, a topology optimization problem is set up, where the maintask is to determine which points in space that should be material pointsand which should be void points.A general structural optimization problem aims to determine the optimalvalue of the design variables such that they maximize or minimize theobjective function while satisfying the constraints. In this case atopology optimization problem is set up with the objective to give thestructure a maximum stiffness, also expressed as minimum compliance givena certain available amount of material volume. The finite elementdiscretized formulation of the topology optimization problems reads:WhereVis the total available volume of materials. See 3 for moredetails in topology optimization. The geometry of the die tool structureis modelled by the use of 3D solid elements. In the set up of a topologyoptimization problem, the total package volume needs to be divided intoa design and a nondesign space, according to the restrictions of theproblem, see Figure 4. Since the objective is to find the optimal materialdistribution in the die tool, without changing the current design of theVolvo S80 truck lid, the material in contact with the blank in the formingprocess are set as a non-design space. That is, the characteristic of thismaterial is not to be changed during the topology optimization. For VCBCto be able to use the same press in manufacturing, the outer shape of thedie is also set as non-design space. To enable the topology optimizationto find the optimal distribution of material, the entire volume underneaththe non-design space is set as available design space, see Figure 4. Thenon-design space is the red area in the figure and the green space, onthe other hand, is the defined design space, which will be optimized.The objective function is set to find the minimum compliance of thepackage domain. The optimal distribution of material in the design domain,given by the internal green elements of the die in Figure 4, is determined.The constraint is set to only use 15% of the volume in the design space,by defining a volume fraction of 0.15. In topology optimization, thestructure is free to take any shape within the given design domain. Oneconcern in topology optimization is that the design concept developed isoften not manufacturable. To satisfy the casting conditions, minimum andmaximum member size of the ribs created can be controlled, as well as thedraw direction of the part. Another feature is pattern grouping thatallows a part of the domain to be designed in a certain pattern, for examplethat two halves of the domain should be symmetrical. The nondesign domain,here represented by red elements, is not a part of the topologyoptimization problem, since it is wished for to remain unchanged.RESULTSRESULTSThe results from the topology optimization can be viewed in Figure 5 andFigure 6. These are the results satisfying the objective function and theconstraints, to minimize the compliance and to use a volume fraction of0.15. The results should be used as an indication of the new design ofthe die for this given load. The outer red frame in the figures is earlierset as non-design space and therefore they are remained fix through thetopology optimization. The red areas in the design space indicate thatthe structure reaches the bolster of the die.A CAD-model is created from the pattern seen in the result of the topologyoptimization. The non-design surfaces are maintained and the new ribstructure is created. Fig 7 and Figure 8 shows the CAD-model of the newoptimized die.When evaluating the result, it is important to keep in mind that thisis the result of an exact given load and that variation in the appliedload might give a different result. A great advantage with topologyoptimizationisthefactthatyouwillgettheoptimaldesign,irrespectively of the number of load cases applied. These load cases canbe of great complexity and hard for the human mind to grasp.DisplacementDisplacement AndAnd StressStress AnalysisAnalysis OfOf OptimizedOptimized DieDieTo analyse the results of the optimized die we look at the displacementsand von Mises stresses in the die. As mentioned earlier, the die issubjected to one forming and one lifting load. The displacements andstresses are remarkably higher in the case of forming compared to thelifting case, and therefore the results from the forming simulation willbe in focus in future discussion. These results will be compared with thedisplacements and von Mises stresses in the original die.DisplacementAnalysisResults of the analyses of the displacements of the optimizedand original die can be viewed in Figure 9 till Figure 12. For thedisplacements to be comparable their directions need to considered.Figure 13 marks the negative direction and magnitude of the displacements.Since they are approximately normal to the surface, both in the originaldie as well as the optimized die, they are comparable. If we focus on theoverall displacements, instead of the maximal displacements, we note thatthe right side of the die in Figure 9 has lower stiffness than the sameside of the VCBC die in Figure 11. This is the result of there being lessmaterial and supporting ribs on this side. The reason for this is thatin the topology optimization the limited material used is better neededon the other side, for this given load. Consideration should naturallybe made that the design is sustainable for variation in applied load.Von Mises Stress AnalysisThe result of the analyse of the von Mises stresses shows that the maximalstress in the optimized die is 140 MPa, and located on the surface of thedie, as shown in Figure 14. In a close up picture of the exposed area,it can be seen that these high stresses acts locally on one element. Sincethe surrounding elements are not nearly as exposed to high stresses, adiscretisation error might have occurred. This discretisation error maybe caused by a defect in the mesh, as well as a numerical problem causedby the mapping procedure. Except for this local error, the overall stressdistribution in the optimized die is comparable to the VCBC die.Table 1 shows the numerical values of the displacements, stresses andweight in the original and optimized die. The weight of the optimized dieis 19%lighter than the original and the maximal displacement is 15% less thesize of the original.CONCLUSIONSCONCLUSIONSThis study show that a die tool of reduced weight and maximum displacementcanbeobtainedthroughtopologyoptimization,giventheloadsestablished inthe forming simulation. An even better new design may beachieved if giving some key aspect some more attention. These include: One limitation is that in the forming simulation the tool parts are setto rigid, which is an approximation to facilitate the calculations. Ifconsidering the actual structural response of the die, a more accurateresult may be obtained from the forming simulation. More efforts can be made to fully describe and understand the loadingcases on the die as well as their relative relevance. For example the loadsfrom the last time step of the forming simulation are used during topologyoptimization as an approximation. A full shape and size optimization should be made in order to get thebest results from the topology optimization. The topology optimizationgives only a coarse layout of the optimal material distribution. Whenadding shape and size optimization to the procedure, the best attainableresult should be found.REFERENCESREFERENCES1. Volvo Car Corporation Body Components,StandardBCD 8203, 004, Castings-Design Instructions(2004).2. F. Birath, A. NilssonTopology Optimization of aStamping Die,Lund: Media-Tryck, 2006.3. M. P. Bendse, O. Sigmund,Topology Optimization:Theory, Methods and Applications,Berlin: Springer-Verlag Heidelberg New York, 2003.冲压模具冲压模具的的拓扑优化拓扑优化AnnaAnna NilssonNilsson andand FridaFrida BirathBirathAnnaNilsson,Altair Engineering 公 司 AB 22370 IDEON 科 技 园 区 ，Scheelevgen15，瑞典 隆德Frida Birath,Epsilon 的高科技创新 AB，Telegrafgatan8A，16984，瑞典 斯德哥尔摩摘要摘要：如今,铸件结构的模具尺寸基本根据标准指南。主要的目标是开发出另一种方法制造冲压工具,它以结构响应的方式考虑生产一个轻量级的设计。目的是通过执行一个拓扑优化来提高结构刚度和减少工具的重量。 模具只是冲压模具考虑方法的其中一部分。 建立的方法可应用于任何其他部分的冲压工具， 同样适用于在负载部分已确定的情况。 两个不同的负载的情况下被施加到模具的工具， 一个操作的情况下和一个运输的情况下， 它们所占比重相等时优化。成形模拟时用 LS Dyna 以获取在操作过程中在模具上的负载。连同模具在运输过程中，使用的软件 OptiStruct 的影响负载，施加到模具和拓扑优化进行时达到的载荷。一个CAD 模型是建立从拓扑优化的结果。分析比较的结果，最大位移，在 Von Mises 应力和模具的重量。最后的结果表明总体比较均匀的应力分布,减少约15%的最大位移和减重约20%。关键词：关键词：有限元法，拓扑优化，板料成型。前言：前言：沃尔沃汽车公司车身部件（从现在开始 VCBC） ，在模具设计和金属板料成形，有丰富的经验， 并且自1920年以来沃尔沃公司一直在声场轿车的车身部分。 多年来所获取的经验有助于我们提高设计工具。今天，该工具的尺寸被引为标准使用， 【1】 。在这些准则的基础上实验 anderror 和结构优化。也因此导致了更高更大强度的需求，这就要求在成型过程中要使用更有强度的配件组成材料。 传统意义上的冲压工具方法是施加共多的材料以维持这一增加的负载，因此能减少一个大的结构响应的风险。然而这种方法并不能无限期地使用下去， 因为在工厂运输中的工具移动式的起重机起重能力是有限的。 因此， 我们考虑到另外的一种方法，制造一种工具，需要考虑到该工具的结构响应，以产生一个轻量的设计方法：方法：现有的冲压工具的制造过程需要在降低其重量的同事考虑到刀具的结构响应。 在下面提出的图1的流程图里。将这种方法应用到沃尔沃 S80车盖成型工具。其中只有一个部件的模具被优化。 请参阅图2.该方法是基于有限元软件HyperWorks的， 使用线性求解器OptiStruct的。 两个不同的负载的情况下被施加到模具的工具， 一个操作的情况下和一个运输的情况下，在优化过程中它们所占比重相等。从成型模拟使用的 LS DYNA 中看出，从最后一个时间不长的仿真模具的接触压力和拉伸力中发现和后面应用中拓扑优化过程中可以实现的操作情况。第二个负载接触了模具的吊耳， 运输和清洁所需的操作。 在拓扑结构优化前提下两个负载的权重相等。然而比解除情况下产生了更大程度的负载，为了安全起见，它们仍需给予相当的关注。金属板材成形模拟金属板材成形模拟：在操作过程中需要将已知作用在模具上的载荷执行拓扑优化。 通过从制造过程中载荷的片材成型模拟得到。 建立 CAD 有限元模型中几何形状的刀具。 该模型和原始工具具有相同的部分。但是冲头，压边和模具是刚性的，在进行坯料接触表面需要模型观察，请参阅图3。进行成型模拟中的详细信息请参加 【2】 。 接触从 LS-DYNA 模型映射到的压力和拉延筋力的有限元模型的拓扑结构优化，通过使用改进的 Delaunay 算法2拓扑优化拓扑优化：为了达到尽可能刚硬的结构， 该方法取决于最优分布在模具内的材料。 为了实现这一目标的拓扑优化问题， 主要的任务是确定空间中无效的材料点。 一般的结构优化的方法是使目标在满足最大化或最小化目标函数的约束条件下，获取设计变量的最优值。在这种情况下，拓扑优化的方法是设立目标， 赋予这个结构一个最大的刚度， 同时材料的可利用量体积遵循最低原则，制定拓扑优化问题的离散有限元写着：其中，V 是可用的材料体积。拓扑优化中的详细信息，参加【3】 。模具工具结构的几何形状是仿造所使用的三维固体元素。在设置的的拓扑优化问题中，根据限制，总包涵量需要被分为一个设计和非设计的空间。请参阅图4、由于我们的目标是在不改变目前设计的沃尔沃 S80卡车盖的前提下，找到模具中最佳材料点，在成型过程中与坯料接触的材料被设置为一个非设计空间内，也就是说此特性材料不会在拓扑优化期间不会被改变。VCBC，可以使用相同的模具， 其外部形状也被设置作为非设计空间。 要启用拓扑优化以找到最佳的材料分布，下面的非设计空间的整个体积都暂时被定义为可用的设计空间， 参见图4.红色区域为非设计空间，绿色区域为定义